Effect of white noise on the \(d\)-dimensional harmonic oscillator. (Effet d’un bruit blanc sur l’oscillateur harmonique de dimension \(d\).) (French) Zbl 0904.60039

Enriquez, Nathanaël

Ann. Inst. Henri Poincaré, Probab. Stat. 32, No. 5, 601-622 (1997).

The author studies the \(d\)-dimensional isotropic harmonic oscillator perturbed by small in some sense random term \(dW_tx_t\), where \(W_t\) denotes the Brownian motion. He obtains explicit formulas for the limit of Lyapunov type exponents \(t^{-1}\ln E(\| x_t\|^2)\) and almost sure limit of \(t^{-1}\ln(\| x_t\|^2+\| v_t\|^2)\), when the small parameter tends to zero.

Reviewer: D.Bobrowski (Poznań)

MSC:

60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
60H30	Applications of stochastic analysis (to PDEs, etc.)
93E15	Stochastic stability in control theory

Keywords:

isotropic harmonic oscillator; Lyapunov exponent; perturbation method